# FOM: cardinals in ZFC fragments

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Wed Aug 7 08:15:53 EDT 2002

> Date: Tue, 6 Aug 2002 15:59:01 -0400
> From: Harvey Friedman <friedman at mbi.math.ohio-state.edu>

> Scott showed that ZF handles cardinals as objects.

The "Hartogs cardinal" of a set X, equal to the set of all
sets Y \in V_a equinumerous to X
--- where a is the least ordinal such that the von Neumann
set V_a contains a set equinumerous to X ---
is a common formal definition of "the cardinal of X" in ZC.
It requires the Foundation Axiom, but does not require AC.

In ZF minus Foundation the Hartogs definition does not
seem to work.
I used to think that it had once established that
ZF minus Foundation just does not admit any universal
definition of "the cardinal of X", but in fact I know no
any reference;
no elementary proof is immediately visible either.

V.Kanovei